*Notes to a video lecture on http://www.unizor.com*

__Superposition of Forces -__

Problems 1

Problems 1

*Problem A*

A boat stands still in the middle of a river. It is tied to a river bank by a rope of the length

**.**

*L*The river flow pushes it with the force

**downstream, and the wind pushes it with the force**

*F***perpendicularly to the river flow away from the bank.**

*0.75·F*(a) What is the force of tension

**of the rope?**

*P*(b) What is the distance

**from the river bank to the boat?**

*d**Answer*:

(a)

*P = 1.25·F*(b)

*d = 0.6·L**Problem B*

An object of mass

**and, therefore, of weight at the ground**

*m***, where**

*W=m·g*

*g*is a free fall acceleration, moves with constant linear speed on a

bridge, that has a shape of an arc with upward convexity of a radius

**.**

*R*(a) Determine the speed

**of an object as a function of pressure**

*v***it produces on the bridge at the top of the arc.**

*P*(b) At what speed

**the object will become "weightless" at the top of the bridge's arc?**

*v*_{0}*Answer*:

(a)

*v = √R·(g − P/m)*(b)

**,**

*v*_{0}= √R·gbecause "weightlessness" is a state when an object does not press on the support and, therefore,

*P=0**Problem C*

A point object of mass

**and, therefore, of weight**

*m***, where**

*W=m·g***is a free fall acceleration, is hanging on the end of a weightless thread of length**

*g***,**

*R*fixed at the other end at some point, and can freely move like a

pendulum within a vertical plane that we can take as the XZ-plane of

coordinates with origin at the point where a thread is fixed.

An object's initial position is with its thread making an angle

**from a vertical.**

*φ*(a) What is the tension of a thread

**in the initial position?**

*P*(b) What is the vector of force

**acting on an object in its initial position in the direction of its motion along a circular trajectory?**

*F**Answer*:

(a) Tension

*P = m·g·cos(φ)*(b) Force along the trajectory

*F = m·g·sin(φ)**Problem D*

A projectile is launched from the wall of the castle horizontally towards the army that laid a siege on this castle.

The height of the castle wall is

**, the horizontal speed of a projectile is**

*H***.**

*v*The free falling acceleration caused by gravity is

**.**

*g*Ignore the air resistance.

(a) What is the time

*the projectile will be in the air until landing?*

**t**_{Land}(b) What is the distance

**of a point the projectile lands from the bottom of the castle wall?**

*d*(c) What is the magnitude of a speed of the projectile

*at the moment of landing?*

**v**_{Land}*Answer*:

(a)

**t**_{Land}=**√2H/g**(b)

**d = v·√2H/g**(c)

**v**_{Land}**= √v² + 2Hg**
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